Theorem ismri2d

Description: Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ismri2.1	|- N = ( mrCls ` A )
	ismri2.2	|- I = ( mrInd ` A )
	ismri2d.3	|- ( ph -> A e. ( Moore ` X ) )
	ismri2d.4	|- ( ph -> S C_ X )
Assertion	ismri2d	|- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ismri2.1	|- N = ( mrCls ` A )
2		ismri2.2	|- I = ( mrInd ` A )
3		ismri2d.3	|- ( ph -> A e. ( Moore ` X ) )
4		ismri2d.4	|- ( ph -> S C_ X )
5	1 2	ismri2	|- ( ( A e. ( Moore ` X ) /\ S C_ X ) -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
6	3 4 5	syl2anc	|- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )